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This paper presents a unified computational framework for the estimation of distances, geodesics and barycenters of merge trees. We extend recent work on the edit distance [106] and introduce a new metric, called the Wasserstein distance between merge trees, which is purposely designed to enable efficient computations of geodesics and barycenters. Specifically, our new distance is strictly equivalent to the L2-Wasserstein distance between extremum persistence diagrams, but it is restricted to a smaller solution space, namely, the space of rooted partial isomorphisms between branch decomposition trees. This enables a simple extension of existing optimization frameworks [112] for geodesics and barycenters from persistence diagrams to merge trees. We introduce a task-based algorithm which can be generically applied to distance, geodesic, barycenter or cluster computation. The task-based nature of our approach enables further accelerations with shared-memory parallelism. Extensive experiments on public ensembles and SciVis contest benchmarks demonstrate the efficiency of our approach -- with barycenter computations in the orders of minutes for the largest examples -- as well as its qualitative ability to generate representative barycenter merge trees, visually summarizing the features of interest found in the ensemble. We show the utility of our contributions with dedicated visualization applications: feature tracking, temporal reduction and ensemble clustering. We provide a lightweight C++ implementation that can be used to reproduce our results.

This paper introduces progressive algorithms for the topological analysis of scalar data. Our approach is based on a hierarchical representation of the input data and the fast identification of topologically invariant vertices, which are vertices that have no impact on the topological description of the data and for which we show that no computation is required as they are introduced in the hierarchy. This enables the definition of efficient coarse-to-fine topological algorithms, which leverage fast update mechanisms for ordinary vertices and avoid computation for the topologically invariant ones. We demonstrate our approach with two examples of topological algorithms (critical point extraction and persistence diagram computation), which generate interpretable outputs upon interruption requests and which progressively refine them otherwise. Experiments on real-life datasets illustrate that our progressive strategy, in addition to the continuous visual feedback it provides, even improves run time performance with regard to non-progressive algorithms and we describe further accelerations with shared-memory parallelism. We illustrate the utility of our approach in batch-mode and interactive setups, where it respectively enables the control of the execution time of complete topological pipelines as well as previews of the topological features found in a dataset, with progressive updates delivered within interactive times., Accepted to IEEE TVCG on February 17th, 2021